To solve this equation, first simplify 4 8/21 + z:

4 8/21 = 4 + 8/21 = 84/21 + 8/21 = 92/21

So, the equation becomes:

(92/21 + z) - 7 13/21 = 2 12/21

Now, simplify the left side of the equation:

(92/21 + z) - 7 13/21 = (92/21 + z) - (147/21) = (92 + 21z - 147)/21 = (21z - 55)/21

Now, set the simplified left side equal to the right side:

(21z - 55)/21 = 2 12/21

To solve for z, first multiply both sides by 21 to get rid of the denominator:

21z - 55 = 2 * 21 + 12
21z - 55 = 42 + 12
21z - 55 = 54

Now, add 55 to both sides:

21z = 54 + 55
21z = 109

Finally, divide by 21 to solve for z:

z = 109/21 = 5 and 4/21

Therefore, z = 5 4/21.